Weighted uniform consistency of kernel density estimators with general bandwidth sequences

Let f(n,h) be a kernel density estimator of a continuous and bounded d-dimensional density f. Let psi(t) be a positive continuous function such that parallel to psi f(beta)parallel to(infinity) < infinity for some 0 < beta < 1/2. We are interested in the rate of consistency of such estimators with respect to the weighted sup-norm determined by psi. This problem has been considered by Gine, Koltchinskii and Zinn ( 2004) for a deterministic bandwidth h(n). We provide "uniform in h" versions of some of their results, allowing us to determine the corresponding rates of consistency for kernel density estimators where the bandwidth sequences may depend on the data and/or the location.